<![CDATA[MKLab - DIFFERENTIAL EQUATIONS]]> https://mklab.gr/ Wed, 28 Oct 2026 04:55:14 +0000 MyBB <![CDATA[Fourier Series & PDEs [Oliver]]]> https://mklab.gr/showthread.php?tid=2002 Tue, 15 Sep 2026 20:46:48 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2002 Fourier Series & PDEs lecture notes provide an undergraduate introduction to partial differential equations, centered on Fourier methods. They begin with Fourier series as a way of representing periodic functions using trigonometric/exponential modes, discussing Fourier coefficients, convergence, differentiation, and approximation in $L^2$. 

The notes then show how Fourier series become a powerful tool for solving classical PDEs through separation of variables, particularly the Laplace, heat, and wave equations. The central strategy is to convert a PDE into simpler ordinary differential equations, determine the resulting coefficients from initial or boundary conditions, and reconstruct the solution as an infinite series. 

The later material develops the Fourier transform for functions on $\mathbb{R}^n$, extending the Fourier-series approach to non-periodic problems. The notes apply these techniques to the Laplace, heat, and wave equations and introduce fundamental solutions, convolution, Green's functions, mean-value formulas, maximum principles, and energy methods. 

A recurring theme is that Fourier analysis transforms differential operators into algebraic or ordinary differential equations in frequency space, making difficult PDE problems tractable. The notes also emphasize qualitative questions such as existence, uniqueness, stability, regularity, and propagation of disturbances, illustrating why PDE theory is important both theoretically and for physical models such as heat flow, vibrating strings, and wave propagation. 

BOOK [PDF]]]> Fourier Series & PDEs lecture notes provide an undergraduate introduction to partial differential equations, centered on Fourier methods. They begin with Fourier series as a way of representing periodic functions using trigonometric/exponential modes, discussing Fourier coefficients, convergence, differentiation, and approximation in $L^2$. 

The notes then show how Fourier series become a powerful tool for solving classical PDEs through separation of variables, particularly the Laplace, heat, and wave equations. The central strategy is to convert a PDE into simpler ordinary differential equations, determine the resulting coefficients from initial or boundary conditions, and reconstruct the solution as an infinite series. 

The later material develops the Fourier transform for functions on $\mathbb{R}^n$, extending the Fourier-series approach to non-periodic problems. The notes apply these techniques to the Laplace, heat, and wave equations and introduce fundamental solutions, convolution, Green's functions, mean-value formulas, maximum principles, and energy methods. 

A recurring theme is that Fourier analysis transforms differential operators into algebraic or ordinary differential equations in frequency space, making difficult PDE problems tractable. The notes also emphasize qualitative questions such as existence, uniqueness, stability, regularity, and propagation of disturbances, illustrating why PDE theory is important both theoretically and for physical models such as heat flow, vibrating strings, and wave propagation. 

BOOK [PDF]]]> <![CDATA[Introduction to Partial Differential Equations [Moore]]]> https://mklab.gr/showthread.php?tid=1956 Sat, 12 Sep 2026 20:11:30 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1956 Book:Introduction to Partial Differential Equations
Author: John Douglas Moore
Date: May 21, 2003
Publisher/Source: Department of Mathematics, University of California

John Douglas Moore’s Introduction to Partial Differential Equations is a concise introductory text designed to build the mathematical machinery needed to understand the principal linear partial differential equations arising in physics and engineering. Moore emphasizes that PDE theory does not revolve around a single universal theorem; instead, different classes of equations require different techniques. The text therefore develops the subject gradually, beginning with power series, ordinary differential equations, eigenvalues, orthogonality, and Fourier analysis before introducing the main PDE models. Physical motivation is central throughout: vibrating strings and membranes, heat diffusion, mechanical systems, and wave propagation provide the intuition behind the mathematics.

The first three chapters establish the analytical foundations. Power series are used to solve differential equations and introduce Bessel's equation, which later becomes essential for studying circular membranes. The book then develops eigenvalues, symmetric matrices, orthonormal bases, and mechanical systems, emphasizing the connection between linear algebra and differential equations. A substantial chapter on Fourier series introduces sine and cosine expansions, inner products, complex Fourier series, and Fourier transforms. These tools form the basic machinery for expressing solutions of boundary- and initial-value problems as infinite sums of eigenfunctions.

The later chapters concentrate directly on PDEs. Moore studies the heat equation, wave equation, and Laplace equation, including initial-value and boundary-value problems, separation of variables, Sturm–Liouville theory, and numerical approximations. The treatment then extends these ideas to higher dimensions, including the Dirichlet problem, the Laplacian in polar coordinates, eigenvalues of the Laplace operator, and the particularly elegant problem of the vibrating circular membrane, where Bessel functions and Fourier analysis come together. The book ends with an appendix showing how Mathematica can be used to solve differential equations.

Main topics
  • Power series and Bessel functions
  • Eigenvalues, eigenvectors, and orthogonality
  • Fourier series and Fourier transforms
  • Heat equation: $u_t=k\Delta u$
  • Wave equation: $u_{tt}=c^2\Delta u$
  • Laplace equation: $\Delta u=0$
  • Separation of variables
  • Sturm–Liouville theory
  • Initial- and boundary-value problems
  • Numerical approximation of PDEs
  • Eigenvalues of the Laplacian
  • Vibrating strings and circular membranes
  • Mathematica applications


BOOK [PDF]]]> Book:Introduction to Partial Differential Equations
Author: John Douglas Moore
Date: May 21, 2003
Publisher/Source: Department of Mathematics, University of California

John Douglas Moore’s Introduction to Partial Differential Equations is a concise introductory text designed to build the mathematical machinery needed to understand the principal linear partial differential equations arising in physics and engineering. Moore emphasizes that PDE theory does not revolve around a single universal theorem; instead, different classes of equations require different techniques. The text therefore develops the subject gradually, beginning with power series, ordinary differential equations, eigenvalues, orthogonality, and Fourier analysis before introducing the main PDE models. Physical motivation is central throughout: vibrating strings and membranes, heat diffusion, mechanical systems, and wave propagation provide the intuition behind the mathematics.

The first three chapters establish the analytical foundations. Power series are used to solve differential equations and introduce Bessel's equation, which later becomes essential for studying circular membranes. The book then develops eigenvalues, symmetric matrices, orthonormal bases, and mechanical systems, emphasizing the connection between linear algebra and differential equations. A substantial chapter on Fourier series introduces sine and cosine expansions, inner products, complex Fourier series, and Fourier transforms. These tools form the basic machinery for expressing solutions of boundary- and initial-value problems as infinite sums of eigenfunctions.

The later chapters concentrate directly on PDEs. Moore studies the heat equation, wave equation, and Laplace equation, including initial-value and boundary-value problems, separation of variables, Sturm–Liouville theory, and numerical approximations. The treatment then extends these ideas to higher dimensions, including the Dirichlet problem, the Laplacian in polar coordinates, eigenvalues of the Laplace operator, and the particularly elegant problem of the vibrating circular membrane, where Bessel functions and Fourier analysis come together. The book ends with an appendix showing how Mathematica can be used to solve differential equations.

Main topics
  • Power series and Bessel functions
  • Eigenvalues, eigenvectors, and orthogonality
  • Fourier series and Fourier transforms
  • Heat equation: $u_t=k\Delta u$
  • Wave equation: $u_{tt}=c^2\Delta u$
  • Laplace equation: $\Delta u=0$
  • Separation of variables
  • Sturm–Liouville theory
  • Initial- and boundary-value problems
  • Numerical approximation of PDEs
  • Eigenvalues of the Laplacian
  • Vibrating strings and circular membranes
  • Mathematica applications


BOOK [PDF]]]> <![CDATA[Introduction to Partial Differential Equations [Herman]]]> https://mklab.gr/showthread.php?tid=1953 Sat, 12 Sep 2026 19:40:11 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1953 Russell L. Herman’s Introduction to Partial Differential Equations is a broad, application-oriented textbook that develops the main analytical and numerical techniques used to study PDEs in mathematics, physics, and engineering. It begins with first-order equations and the method of characteristics, including conservation laws, shock and rarefaction waves, then introduces the classical second-order heat, wave, and Laplace equations through boundary-value problems, separation of variables, and d’Alembert’s solution. 

The book progressively builds the mathematical machinery needed for more advanced problems, covering Fourier series, Sturm–Liouville theory, Legendre and Bessel functions, higher-dimensional equations, spherical harmonics, Green’s functions, complex-variable methods, Fourier and Laplace transforms, and finally finite-difference numerical methods with discussions of stability and error. Throughout, theory is tied to concrete models such as vibrating strings and membranes, heat flow, fluid flow, traffic flow, quantum mechanics, and diffusion, making the text essentially a comprehensive two-semester introduction to both the theory and practical solution of partial differential equations. 

BOOK [PDF]]]> Russell L. Herman’s Introduction to Partial Differential Equations is a broad, application-oriented textbook that develops the main analytical and numerical techniques used to study PDEs in mathematics, physics, and engineering. It begins with first-order equations and the method of characteristics, including conservation laws, shock and rarefaction waves, then introduces the classical second-order heat, wave, and Laplace equations through boundary-value problems, separation of variables, and d’Alembert’s solution. 

The book progressively builds the mathematical machinery needed for more advanced problems, covering Fourier series, Sturm–Liouville theory, Legendre and Bessel functions, higher-dimensional equations, spherical harmonics, Green’s functions, complex-variable methods, Fourier and Laplace transforms, and finally finite-difference numerical methods with discussions of stability and error. Throughout, theory is tied to concrete models such as vibrating strings and membranes, heat flow, fluid flow, traffic flow, quantum mechanics, and diffusion, making the text essentially a comprehensive two-semester introduction to both the theory and practical solution of partial differential equations. 

BOOK [PDF]]]> <![CDATA[Partial Differential Equations [Ivrii]]]> https://mklab.gr/showthread.php?tid=1952 Sat, 12 Sep 2026 19:34:33 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1952 Partial Differential Equations is a comprehensive, freely available undergraduate textbook based on the University of Toronto’s APM346 course, introducing the main ideas and techniques of PDEs through both theory and applications. It begins with PDE classification, initial and boundary value problems, and physical motivations, then develops the fundamental wave, heat, and Laplace equations, using characteristic methods, energy arguments, separation of variables, Fourier series and Fourier transforms, eigenvalue problems, Green’s functions, and potential theory. 
Later chapters move into more advanced material including variational methods, distributions and weak solutions, nonlinear equations such as Burgers and Navier–Stokes, spectral theory, scattering, conservation laws, Maxwell equations, and quantum-mechanical operators. Throughout, the book combines derivations, examples, applications, and substantial problem sets, making it suitable both as a course text and as a broad reference for learning the classical analytical toolkit of partial differential equations. 

BOOK [PDF]]]> Partial Differential Equations is a comprehensive, freely available undergraduate textbook based on the University of Toronto’s APM346 course, introducing the main ideas and techniques of PDEs through both theory and applications. It begins with PDE classification, initial and boundary value problems, and physical motivations, then develops the fundamental wave, heat, and Laplace equations, using characteristic methods, energy arguments, separation of variables, Fourier series and Fourier transforms, eigenvalue problems, Green’s functions, and potential theory. 
Later chapters move into more advanced material including variational methods, distributions and weak solutions, nonlinear equations such as Burgers and Navier–Stokes, spectral theory, scattering, conservation laws, Maxwell equations, and quantum-mechanical operators. Throughout, the book combines derivations, examples, applications, and substantial problem sets, making it suitable both as a course text and as a broad reference for learning the classical analytical toolkit of partial differential equations. 

BOOK [PDF]]]> <![CDATA[Differential Equations for Engineers [Lebl]]]> https://mklab.gr/showthread.php?tid=1095 Mon, 13 Jul 2026 18:13:45 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1095 Differential Equations for Engineers
By  Jiří Lebl
Summary
Notes on Diffy Qs: Differential Equations for Engineers by Jiří Lebl is a free, beginner-friendly textbook that introduces ordinary differential equations with a strong emphasis on understanding concepts rather than simply memorizing solution techniques. Written for students who have completed basic calculus, it combines clear explanations with practical examples drawn from engineering, physics, and other scientific fields. 
The book covers the essential topics of a first differential equations course, including first- and second-order equations, systems of differential equations, Laplace transforms, power series, Fourier methods, and an introduction to partial differential equations. 
Throughout the text, mathematical theory is balanced with real-world applications, helping readers see why differential equations are such powerful tools for modeling natural phenomena. Numerous worked examples, exercises, and intuitive explanations make the material approachable for self-study as well as classroom use. As an open educational resource, the book is freely available online and has been widely adopted by instructors seeking an accessible yet comprehensive introduction to differential equations. 
BOOK
]]> Differential Equations for Engineers
By  Jiří Lebl
Summary
Notes on Diffy Qs: Differential Equations for Engineers by Jiří Lebl is a free, beginner-friendly textbook that introduces ordinary differential equations with a strong emphasis on understanding concepts rather than simply memorizing solution techniques. Written for students who have completed basic calculus, it combines clear explanations with practical examples drawn from engineering, physics, and other scientific fields. 
The book covers the essential topics of a first differential equations course, including first- and second-order equations, systems of differential equations, Laplace transforms, power series, Fourier methods, and an introduction to partial differential equations. 
Throughout the text, mathematical theory is balanced with real-world applications, helping readers see why differential equations are such powerful tools for modeling natural phenomena. Numerous worked examples, exercises, and intuitive explanations make the material approachable for self-study as well as classroom use. As an open educational resource, the book is freely available online and has been widely adopted by instructors seeking an accessible yet comprehensive introduction to differential equations. 
BOOK
]]> <![CDATA[Elementary Differential Equations [Trench]]]> https://mklab.gr/showthread.php?tid=250 Wed, 10 Jun 2026 03:41:49 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=250 William F. Trench’s Elementary Differential Equations with Boundary Value Problems is a comprehensive introductory textbook aimed at undergraduate students in mathematics, engineering, and the physical sciences. It develops differential equations from the basic concepts and first-order equations through numerical methods, applications, linear second- and higher-order equations, series solutions, Laplace transforms, and systems of differential equations. The presentation is deliberately conversational and student-friendly, with more than 250 worked examples and a very large collection of exercises ranging from routine calculations to challenging problems. 

The later chapters move toward more advanced applications, particularly boundary-value problems, Fourier series, and partial differential equations. Trench emphasizes both analytical techniques and computational/graphical approaches, including exercises designed for technology-assisted exploration. A major strength is the balance between conceptual explanation and practical problem solving: rather than overwhelming the reader with formal theorems, the book builds intuition through examples, figures, applications, and carefully graded exercises. Overall, it is an excellent self-study or teaching text for developing a solid undergraduate foundation in ordinary differential equations and boundary-value problems.


BOOK PAGE]]> William F. Trench’s Elementary Differential Equations with Boundary Value Problems is a comprehensive introductory textbook aimed at undergraduate students in mathematics, engineering, and the physical sciences. It develops differential equations from the basic concepts and first-order equations through numerical methods, applications, linear second- and higher-order equations, series solutions, Laplace transforms, and systems of differential equations. The presentation is deliberately conversational and student-friendly, with more than 250 worked examples and a very large collection of exercises ranging from routine calculations to challenging problems. 

The later chapters move toward more advanced applications, particularly boundary-value problems, Fourier series, and partial differential equations. Trench emphasizes both analytical techniques and computational/graphical approaches, including exercises designed for technology-assisted exploration. A major strength is the balance between conceptual explanation and practical problem solving: rather than overwhelming the reader with formal theorems, the book builds intuition through examples, figures, applications, and carefully graded exercises. Overall, it is an excellent self-study or teaching text for developing a solid undergraduate foundation in ordinary differential equations and boundary-value problems.


BOOK PAGE]]> <![CDATA[Ordinary Differential Equations and Dynamical Systems [Teschl]]]> https://mklab.gr/showthread.php?tid=232 Wed, 10 Jun 2026 02:43:34 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=232
Ordinary Differential Equations and Dynamical Systems
Gerald Teschl

Summary  This book provides an introduction to ordinary differential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental results concerning the initial value problem: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore we consider linear equations, the Floquet theorem, and the autonomous linear flow.
Then we establish the Frobenius method for linear equations in the complex domain and investigate Sturm-Liouville type boundary value problems including oscillation theory.
Next we introduce the concept of a dynamical system and discuss stability including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems.
We prove the Poincare-Bendixson theorem and investigate several examples of planar systems from classical mechanics, ecology, and electrical engineering. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed as well.
Finally, there is an introduction to chaos. Beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits.


BOOK PAGE
]]>
Ordinary Differential Equations and Dynamical Systems
Gerald Teschl

Summary  This book provides an introduction to ordinary differential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental results concerning the initial value problem: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore we consider linear equations, the Floquet theorem, and the autonomous linear flow.
Then we establish the Frobenius method for linear equations in the complex domain and investigate Sturm-Liouville type boundary value problems including oscillation theory.
Next we introduce the concept of a dynamical system and discuss stability including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems.
We prove the Poincare-Bendixson theorem and investigate several examples of planar systems from classical mechanics, ecology, and electrical engineering. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed as well.
Finally, there is an introduction to chaos. Beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits.


BOOK PAGE
]]> <![CDATA[Linear Partial Differential Equations and Fourier Theory [Pivato]]]> https://mklab.gr/showthread.php?tid=226 Wed, 10 Jun 2026 02:22:29 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=226 Linear Partial Differential Equations and Fourier Theory
by Marcus Pivato

Summary  This is a textbook for an introductory course on linear partial differential equations and initial/boundary value problems. It also provides a mathematically rigorous introduction to basic Fourier analysis, which is the main tool used to solve linear PDEs in Cartesian coordinates. Finally, it introduces basic functional analysis. This is necessary to rigorously characterize the convergence of Fourier series, and also to discuss eigenfunctions for linear differential operators.

BOOK PAGE]]> Linear Partial Differential Equations and Fourier Theory
by Marcus Pivato

Summary  This is a textbook for an introductory course on linear partial differential equations and initial/boundary value problems. It also provides a mathematically rigorous introduction to basic Fourier analysis, which is the main tool used to solve linear PDEs in Cartesian coordinates. Finally, it introduces basic functional analysis. This is necessary to rigorously characterize the convergence of Fourier series, and also to discuss eigenfunctions for linear differential operators.

BOOK PAGE]]> <![CDATA[Finite Difference Computing with PDEs [Langtangen]]]> https://mklab.gr/showthread.php?tid=215 Wed, 10 Jun 2026 01:42:18 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=215 Finite Difference Computing with PDEs
by Hans Petter Langtangen, Svein Linge

Summary

Finite Difference Computing with PDEs: A Modern Software Approach by Hans Petter Langtangen and Svein Linge is a practical, open-access textbook that introduces numerical methods for solving partial differential equations (PDEs) using finite-difference techniques and computer programming. 

Rather than focusing only on mathematical theory, it guides readers from the formulation of numerical schemes through algorithm design, Python implementation, verification, and interpretation of simulation results, using examples involving vibration, wave, diffusion, advection-dominated, and nonlinear equations. 

The book emphasizes understanding both the mathematical behavior and computational artifacts of numerical solutions, with applications particularly relevant to physics, engineering, and biology. 

BOOK PAGE [FREE AS A PDF]]]> Finite Difference Computing with PDEs
by Hans Petter Langtangen, Svein Linge

Summary

Finite Difference Computing with PDEs: A Modern Software Approach by Hans Petter Langtangen and Svein Linge is a practical, open-access textbook that introduces numerical methods for solving partial differential equations (PDEs) using finite-difference techniques and computer programming. 

Rather than focusing only on mathematical theory, it guides readers from the formulation of numerical schemes through algorithm design, Python implementation, verification, and interpretation of simulation results, using examples involving vibration, wave, diffusion, advection-dominated, and nonlinear equations. 

The book emphasizes understanding both the mathematical behavior and computational artifacts of numerical solutions, with applications particularly relevant to physics, engineering, and biology. 

BOOK PAGE [FREE AS A PDF]]]> <![CDATA[A First Course in Elementary Differential Equations [Finan]]]> https://mklab.gr/showthread.php?tid=214 Wed, 10 Jun 2026 01:40:07 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=214 A First Course in Elementary Differential Equations
by Marcel B. Finan

Summary  This note covers the following topics: Qualitative Analysis, Existence and Uniqueness of Solutions to First Order Linear IVP, Solving First Order Linear Homogeneous DE, Solving First Order Linear Non Homogeneous DE: The Method of Integrating Factor, Modeling with First  Order Linear Differential Equations.

BOOK PAGE]]> A First Course in Elementary Differential Equations
by Marcel B. Finan

Summary  This note covers the following topics: Qualitative Analysis, Existence and Uniqueness of Solutions to First Order Linear IVP, Solving First Order Linear Homogeneous DE, Solving First Order Linear Non Homogeneous DE: The Method of Integrating Factor, Modeling with First  Order Linear Differential Equations.

BOOK PAGE]]>